We introduce the Hierarchically Growing Hyperbolic Self-Organizing Map (H2SOM) featuring two extensions of the HSOM (hyperbolic SOM): (i) a hierarchically growing variant that allows for incremental training with an automated adaptation of lattice size to achieve a prescribed quantization error and (ii) an approximate best match search that utilizes the special structure of the hyperbolic lattice to achieve a tremendous speed-up for large map sizes. Using the MNIST and the Reuters-21578 database as benchmark datasets, we show that the H2SOM yields a highly efficient visualization algorithm that combines the virtues of the SOM with extremely rapid training and low quantization and classification errors.